Apparatus for determining the magnitudes required for parallax correction



I SR 5 x QR 2 s 3 7 6 a 71 9 f May 22, 1945.- K. PAPELLO 7 APPARATUS FOR DETERMINING THE MAGNITUDES REQUIRED FOR PARALLAX CORRECTION Filegi June 7, 1938 3 Sheets-Sheet 1 Invenipr: M MAM y n L H HWUWE 23b. mmswfisa 5 i. Q g

May 22, 1945. K. PAPELLO 2,376,719

APPARATUS FOR DETERMINING THE MAGNITUDES REQUIRED FOR PARALLAX CORRECTION Flled June 7, 1938 3 Sheets-Sheet 2 IIIIIHIllllllllllllilllllllllilll 7 l'fl= 11/! m..

Inventor."

M Mu,

235. REGkSTE'WQ-i May 22, 1945. K. PAPELLO APPARATUS FOR DETERMINING THE MAGNITUDES REQUIRED FOR PARALLAX CORRECTION 5 Sheets-Sheet 3 Filed June 7, 193

lnyenlor: \KQA Wm ncrno l and" Patented May 22, 1945 gun-ti 1) APPARATUS FOR DETERMINING THE MAG- NITUDJES REQUIRED FOR P BALLAX CORRECTION Karl Papello, Jena, Germany; vested in the Alien Property Custodian Application June 7, 1938, Serial No. 212,349 In Germany May 4, 1934 9 Claims.

The invention refers to an apparatus in which the parallax correction required for laying an anti-aircraft gun positioned at a certain distance from the apparatus is determined by means of the range E of the aircraft from the apparatus, the horizontal distance 8 of the gun from the apparatus, the lateral angle at between the lines of sight from the apparatus to the aircraft and to the gun, and theelevational angle of the line of sight from the apparatus to the aircraft.

The invention aims at considerably simplifying the construction of such apparatus by providing three cam systems each of which comprises a plurality of cams arranged in series or trains and each effecting two different motions, for instance a rotation and. a displacement, and so constructed and connected to each other and to driving mechanisms feeding the known magnitudes E, s, on and 13 that the rearmost cams of the three systems furnish respectively the difference, Act of the lateral angles included between a definite reference line and the lines of sight from the apparatus and the gun to the aircraft, the difference A5 of the elevational angles of the lines of sight from the apparatus and the gun to the aircraft, and the difference AE of the ranges of the aircraft from the apparatus and the gun, the said magnitudes Act, A5 and AE corresponding to the equations s.sin a (EiAE).cos (dint) s.cos (uni?) sin ,6

A [3 arcsin EiAE (I) Aa=arosin and (III) AE=s.cos ad= )-cos (5i?) I in Equation III.

In the accompanying drawings, Figures 1 and 2 show schematically how the equations determining the sought magnitudes Au, Ad and AE are deduced, and Figures 3 and 4 illustrate in schematic plan Views two constructional examples of an apparatus according to the invention.

In Figures 1 and 2, A is the apparatus, B the gun, and H the aircraft. For the sake of simplicity, A and B are assumed to lie in a horizontal plane. C is the projection of H in this plane. D is an auxiliary point lying in this plane and so positioned that C'D=AB=s and F is the point at which a perpendicular dropped from D intersects AC, and G is a point lying in AC and whose distance AG from A is equal to AD. The angle CAD is equal to the angle ACE and illustrative of the difference Ad of the lateral angles included between a definite reference line, i. e., the line AB, and the lines of sight from the apparatus A and the gun B to the aircraft H. The angle FDG is half the angle CAD and therefore equal to J is a point vertically above G and lying in the horizontal plane containing H, K is the point at which AH is intersected by a perpendicular dropped from J, and L is a point which lies in AH and whose distance AL from A is equal to AJ. AJ=AL is equalto the range E0 of the aircraft H from the gun B, and AH=E is the range of the aircraft H from the apparatus A, so that LH is the difierence AE of the said ranges E and E0. The angle CAH, viz. the elevational angle of the line of sight from the apparatus A to the aircraft H, is designated ,8. p is also the magnitude of the angle JHA. The angle GAJ is equal to the elevational angle [30 of the line of sight from the gun B to the aircraft H. The angle HAJ=Ap determines the diflference between these elevational angles 5 and B0. The angle KJL is half the angle HAJ and, accordingly,

It follows at once from Figures 1 and 2, due regard being had to the necessary generalizations, that s.sin 0:

(I) Aa=arcsin FG=s sin a tan and that, accordingly,

AOL JH=s(cos as1n tan Assuming in the case of small angles that and Au: Aa tan Y-Sll'l i there is obtained JH=s cos 11 cos %=Fsin a sin g):

s cos (at?) sin Afi= cos (calf sin B and, accordingly,

s cos (011%) sin 6 EiAE AE=HL=HKKL (II) AB arcsin HK=JH cos B=s cos (ai )-cos B KL=JK tan A2B=JH sin a tan s cos (Q1 sin B tan Assuming in the case of small angles that fit cos 2 1 and sin 92 tan 2 there is obtained AE=s cos (azb cos [3.00s A6 AB 2 ism B SlIl (III) AE=s cos (ai?) cos r g) which differ from the Equations I and II only in that the sines of A0: and A3 are considered to be equal to the magnitude of the angle itself, as is admissible when small angles are concerned. If the demands made upon the apparatus are reduced even further, the determination of As, no and AE can be based on the equations These equations follow from the Equations Ia, 11a and III by neglection of AE and A5 in la, AE and Am in 11a, and Au and A6 in III.

The apparatus illustrated by Figure 3 is used for the determination of the magnitudes A, A5 and AE according to the Equations Ia, 11a and III. On a plate I are rotatably mounted three shafts 2, 3 and 4, which are assumed to be continuously rotated by other parts (not represented in the drawings) of the gunnery calculator according to the angle a, the angle 18 and the range E, respectively. By means of a hand wheel 5 and a threaded spindle 6, a slide 1 can be adjusted according to the horizontal distance s of the gun from the apparatus. The correponding indicating device is designated 8, 9. The shaft 3, actuated according to B, rotates by means of a pair of spur gear wheels II], II and a shaft l2 the crown wheel l3 of a differential wheel gear I 4 whose planet wheels I5 are rotated by a shaft l6 (operated according to A3, as explained hereinafter) and a pair of spur gear wheels |8 about the axis of the shaft l2 according to so that the other crown wheel IQ of the differential wheel gear l4 and the shaft 20 fast therewith are rotated according to fio=fii,8. The shaft 4 actuated according to E operates by means of a pair of spur gear wheels 2|, 22 the planet wheels 23 of a differential wheel gear 24 according to E/2. The one crown wheel 25 of this gear 24 is rotated by a shaft 26 (operated accordin to AE, as explained hereinafter) and a. pair of bevel gear Wheels 21 according to AE, s that the other crown wheel 28 of the differential wheel gear 24 and the shaft 29 fast therewith are rotated according to Eo=E :AE.

For finding All use is made of a cam system comprising three cams 30, 3| and 32. The cam 38 is displaceable along a grooved shaft 33 rotatable according to E0 by means of the shaft 29 and a pair of bevel gear wheels 34. The slide 1 engages by means of a fork 35 the nave 36 of the cam 30 in such a manner that this cam participates in the displacements 0f slide 1. The cam 3|], which is thus displaced according to s and rotated according to E0, is so constructed that a hollow feeler 31 displaceable along a grooved shaft 38 and continuously kept in contact with the cam 38 by means of a spring 39 is adjusted according to constructed that a hollow feeler 4| displaceable along a grooved shaft 42 and continuously kept ice: iii: iiiiiiii in contact with the cam 3| by means of a spring 43 is adjusted according to sin a The shaft 42 is coupled by means of a pair of bevel gear wheels 44 to the shaft 26 and, therefore, rotated according to filo. The cam 32, which is fast with the feeler 4| and therefore displaced according to E 5111 Ct and rotated according to fin, is so constructed that a feeler 45 mounted on the plate I for displacement at rignt angles to the shaft 42 and continuously kept in contact with the cam 32 by means of a spring 46 is adjusted according to viz. according to the sought magnitude Aa. The feeler 45 is fast with a rack 41 in mesh with a toothed wheel 48. By means of two pairs of bevel gear wheels 49 and 50, the rotations of the toothed wheel 48 are transmitted to a shaft which is therefore rotated according to Air. From this shaft 5| the said magnitude Act can be transmitted to other mechanisms of the firing director which are not shown in the drawings.

The shaft 38 rotated according to a carries the one crown wheel 52 of a differential wheel gear 53, whose planet wheels 54 are coupled to the shaft 5| by means of two pairs of bevel gear wheels 55 and 56 in such a manner as to be rotated about the axis of the shaft 38 according to so that the other crown wheel 51 of the differential wheel gear 53 and the grooved shaft 58 carrying this crown wheel 51 are rotated according to Ace ad; 2

For finding A18, use is made of a cam system comprising three cams 59, 6|] and 6|. The cam 59 is displaceable along the grooved shaft 58. By means of a pair of bevel gear wheels 62 a threaded spindle 63 parallel to the shaft 58 is coupled to the threaded spindle 6 in such a manner that a slide 64 displaced by the spindle 63 is adjusted according to s. The slide 64 engages by means of a fork 65 the nave '66 of the cam 59 in such a manner that this cam 59 participates in the displacements of the slide 64. The cam 59, which is thus displaced according to s and rotated according to is so constructed that a feeler 61 mounted on the plate I for displacement at right angles to the shaft 58 and continuously kept in contact with the cam 59 by means of a spring 68 is adjusted according to s cos (azlz The cam 60 is displaceable along a grooved shaft 69 which is parallel to the feeler 61 and rotated I by the shaft l2 and a pair of bevel gear wheels 19 according to ,3. By means of a fork 1|, the feeler 61 engages the nave 12 of the cam 6|) in such a manner that this cam66 participates in the adjustments of the feeler 61. The cam 6|), which is thus displaced according to s cos (out?) and rotated according to p is so constructed that a hollow feeler 13 displaceable along a, grooved shaft 14 and continuously kept in contact with the cam 60 by means of a spring 15 is adjusted according to s cos ((11%) sin [3 By means of a pair of bevel gear wheels 16, a shaft 11, a pair of bevel gear wheels 18, a shaft 1.9 and a pair of bevel gear wheels 80, the shaft 14 is coupled to the shaft 29 in such a manner as to be rotated, as is the shaft 29, according to E0. The cam 6|, which i fast with the feeler 13 and therefore displaced according to 8 cos (mi sin B and rotated according to E0, is so constructed that a feeler 8| mounted on the plate I for displacement at right angles to the shaft 14 and continuously kept in contact with the cam 6| by means of a spring 82 is displaced according to cos (iii sin 3 viz. according to the sought magnitude A5. The feeler 8| is fast with a rack 83 in mesh with a toothed wheel 84. By means of a pair of bevel gear wheel 85 the rotations of the toothed wheel 84 are transmitted to the shaft |6, so that, as assumed above, this shaft I6 is rotated according to As. From the shaft l6, the magnitude A}? can be transmitted to other mechanisms of the firing director which are not represented in the drawings.

The shaft |2 rotated according to p? carries the one crown wheel 86 of a differential wheel gear 81 whose planet wheels 88 are coupled by means of a pair of spur gear wheels 89 to the shaft I6 in such a manner as to rotate about the axis of the shaft I2 according to 4 so that the other crown wheel 99 of the differential wheel gear 81 is rotated according to By means of a pair of bevel wheels 9| a grooved shaft 92 is coupled to the crown wheel 90 in such a manner as to rotate likewise according to Along the shaft 92 is displaceable a cam 93 constituting together with the cam 59 the third cam system, which serves for finding AE. The feeler 61 engages by means of a fork 95 the nave 94 of this cam 93 in such a manner that this cam 93 participates in the displacements of the feeler 61. The cam 93, which is thus displaced accord ing to and rotated according to i so constructed that a feeler 96 mounted on the plate I for displacement at right angles to the shaft 92 and continuously kept in contact with the cam 93 by means of a spring 91 is adjusted according to 3 cos (we?) cos (8.45%

viz. according to the sought magnitude AB. The feeler 96 is fast with a rack 98 in mesh with a toothed wheel 99. By means of a pair of bevel gear wheels I the rotations of the toothed wheel 99 are transmitted to the shaft 26, so that, as assumed above, this shaft 26 is rotated according to AE. From the shaft 26, the magnitude AE can be transmitted to other mechanisms of the firing director which are not represented in the drawings.

The described apparatus works automatically as soon as the slide I is adjusted by means of the hand wheel according to s and the shafts 2, 3 and 4 are rotated according to on, s and E, respectively.

An apparatus for maintaining the Equations I, II and III would differ from the described constructional example only by a different construction of the cams 32 and BI. The cam 32 would have to be so constructed as to adjust the feeler 45 according to s sin a s cos (ai sin [5 AB arcsin E0 when displaced according to s cos (a 1%) sin 6 and rotated according to E0.

The apparatus illustrated by Figure 4 serves for finding the magnitudes Am, As and AE according to the Equations Ib, IIb and IIIb. 0n account of the smaller demands made upon the accuracy of the found magnitudes, this apparatus is more simple than that shown by Figure 3, the main difference being that it lacks any differential wheel gears. In the following description, the designations are as far as possible the same as in Figure 3.

The mechanisms for calculating Aa differ from those of the apparatus shown by Figure 3 only in so far as the grooved shaft 33 is coupled by means of a pair of bevel gear wheels 34 direct to the shaft 4 operated according to E and is therefore rotated according to E (and not according to E0), and in that the hollow feeler 31 is displaceable along the grooved shaft 2 rotated according to a, and in that the shaft 42 is coupled by means of the pair of bevel gear wheels 44 direct to the shaft 3 operated according to p and is therefore rotated according to ,6 (and not according to flu). The feeler 45 is consequently displaced according to (and not according to s sin or) E cos B The transformation of this displacement into a rotation is effected by means of a follower mechanism. This mechanism has a contact piece IOI fast with the feeler 45 and constituting together with two contact members I02 and I03 an electric contact device. The two contact members I02 and I03 are connected to a slide I04 which is displaceable in the one or in the other direction by means of a threaded spindle I05, a pair of spur gear wheels I06 and a motor I0'I' when the contact piece IOI touches the contact member I02 or I03. From the spindle I05, the magnitude Act can be transmitted to other mechanisms of the firing director which are not represented in the drawings.

The cam system for finding As comprises two cams I08 and I09. The cam I00, which is rigidly connected to the cam 3I and therefore likewise displaceable according to E/s and rotatable according to a, is so constructed that a hollow feeler IIO displaceable along a, grooved shaft III and continuously kept in contact with the cam I08 by means of a spring H2 is adjusted according to i COSoz The shaft III is coupled by means of a pair of bevel gear wheels II3 to the shaft 3 in such a manner as to be rotated according to ,8. The cam I09, which is fast with the feeler H0 and therefore displaced according to and rotated according to c, is so constructed that a feeler II4 mounted on the plate I for displacement at right angles to the shaft III and continuously kept in contact with the cam I09 by means of a spring H5 is displaced according to 8 cos sin 5 a E viz. according to the sought magnitude AB. The transformation of this displacement into a rotation is effected by means of a follower mechanism comprising a contact piece I I6 which is fast with thefeeler H4 and constitutes together with two contact members Ill and H8 an electric contact device. The two contact members III and H8 are connected to a slide II9 which is displaceable in the one or in the other direction by means of a threaded spindle I20 and a pair of spur gear wheels I2I and a motor I22 when the contact piece II6 touches the contact member I" or II8. From the spindle I20, the magnitude As can be transmitted to other mechanisms of the firing director which are not represented in the drawings.

Of the two mechanisms 59 and 93 for finding AE, 59 is displaceably mounted on the grooved shaft 2 rotated according to 0:. This mechanism 59 is therefore not only displaced according to s but also rotated according to a. The cam 59 is so constructed that a hollow feeler I23 displaceable along a grooved shaft I24 and continuously kept in contact with the cam 59 by means of a spring I25 is adjusted according to s cos oz. The shaft I24 is so coupled by means of a pair of bevel gear wheels I26 to the shaft 3 as to rotate according to p. The cam 93, which is fast with the feeler I23 and therefore displaceable according to 8 cos a and rotatable according to p, is so constructed that a pin I2I mounted on the plate ECHO I Lnwv I for displacement at right angles to the shaft I24 and kept in continuous contact with the cam 93 by means of a spring I28 is displaced according to s cos a cos )3, viz. according to the sought ma 3. In an apparatus comprising a system of three multiplication devices according to claim 1 and serving for determining the three magnitudes for parallax correction required for laynitude AE. The transformation of this displace- 5 ing an anti-aihcraft gun positioned at a cerment into arotation is effected by means of a tain distance from the apparatus, viz. the corfollower mechanism. This mechanism has a conrection of the lateral angle Aer, the correction of tact piece I29 fast with the pin I21 and constithe elevational angle A5 and the correction of tuting together with two contact members I30 the range AE, each of said three magnitudes beand I3I an electric contact device. The two coning the product of at least three factors contact members I30 and I3I are connected to a stituting functions of the following given values: slide I32 which is displaced in the one or in the the range E of the aircraft from the apparatus, other direction by means of a threaded spindle the horizontal distance s of the gun from the ap- I33, a pair of spur gear wheels I34 and a motor paratus, the lateral angle or between the lines of I35 when the contact piece I29 touches the con- Sight from e pp at to e a t and to tact member I30 or I 3|. From the spindle I33 the gun, and the elevatinoal angle ,8 of the line the magnitude AE can be transmitted to other of sight fromv the apparatus to the aircraft, ,mechanisms of the firing director which are not means for coupling said three devices relatively represented in the drawings. to each other, said means comprising differential In deducing the equations for Au, A5 and AE, gears for imparting to at least one cam of each it has been assumed for the sake of simplicity said device an additional motion, and further that the apparatus A and the gun B lie in a horimeans for .coupling the gearing of the difierzontal plane. Taking into consideration any difential ear of ea h a d device to the feeler of ference in the heights of A and B does not prethe rearmost cam of another device. sent any difliculties either in the deduction of 5 An app ratu comprising h ee y te s of the corresponding equations or in the constructrains of two-dimensional cams as set forth in tion of a corresponding apparatus, Iclaim 1 for determining the three magnitudes for I claim: parallax correction required for laying an anti- 1, In an apparatus for determining t instam aircraft gun positioned at a distance from the taneous product of at least three factors consti- Point of observation of the target, namely, the tuting functions of variable magnitudes, a plucorrection 0f the lateral angle the o e rality of multiplication devices each having at tich cf the elevational angle 6 a t correcleast two two-dimensional cams connected for tion of the range E as Observed to correspond operation in series in each such device and each to the lateral angle, elevational angle and range, of which cams is shaped for producing the prodrespectively, at the gun position, Said sy te s ni; of t f t feeler for each of Said cluding means for introducing factors representcams cooperating therewith, means fo coupling ing the observed lateral angle, elevational angle the feeler'for the foremost cam of each multiand range, respectively, each to e difierent one of plication device to the following cam thereof for Said Systems of multiplying Cams, and gear imparting th t a ti corresponding t t 40 means associated therewith for combining the product produced by said foremost cam, means Several end products of the multiplying Cams of for imparting t id following cam a motion the several systems thereof with said factors corresponding to a third factor, and means for nt n th erv la a n v coupling said multiplication devices to combine ticnel angle and range, respectivelythe respective products thereof whereby a change An apparatus ccmplising t ee syste s of in any one" of said magnitudes is reflected in said trains of two-dimensional calm as Set ferth in combined products. claim 1 for determining the three magnitudes for 2. In an apparatus comprising a system of parallax correction required for laying an antithree multiplication devices according to claim 1 r a t gun positioned t a tan e f om t e and serving for determining the three magpoint of observation of the target, namely, the nitudes for parallax correction required for laycorrection of the lateral angle a, the correction ingan anti-aircraft gun positioned at a certain of the elevational angle 13 and the correction of distance from the apparatus, viz, the correction the range E as bse ved to rr p nd to the latof the lateral angle Ana, the correction of the eleeral ang e, elevational angle and range, e p vational angle AB and the correction of the range t vely, at the n p s t said Systems includ- AE; each of said three magnitudes being the ing means f r in r in fa t p entin product of at least three factors constituting the observed lateral angle, elevational angle functions of the following given values: the range and range, respe tive ea h to a different one E of the aircraft from the apparatus, the hori-- of said systems of multiplying cams, gear means zontal distance s of the gun from the apparafor combining the several end products of the tus, the lateral angle or between the lines of sight multiplying cams of the several systems thereof from the apparatus to the aircraft and to the with said factors representing the observed latgun, and the elevational angle p of the line of oral angle, elevational angle and range, respecsight from the apparatus to the aircraft, means tively, and means for simultaneously adjusting for coupling said three devices relatively to each one of the cams of each of said systems to corother, said coupling means comprising means respond to changes in the distance of the gun for so coupling one cam of one of said three defrom the point of observation. vices to one cam of another device that the two 6. An apparatus for determining the magmotions of the one cam correspond to the two nitudes for parallax correction required for laymotions of the other cam, respectively, and coming an anti-aircraft gun positioned at a distance prising further means for so coupling one cam therefrom for firing at a moving target overhead, of one of said three devices to one cam of the secsaid apparatus comprising four independently 0nd device and to one cam of the third device operatable coordinated mechanisms for the sithat said three cams have one of their two multaneous bringing together of four magnitudes motions in common. 7 respectively representing the horizontal distance S of the gun from said apparatus, the range E of the target from said apparatus at a certain time, the lateral angle a between the line of sight from said apparatus to the target and the line of sight from said apparatus to the gun at the same time, and the elevational angle 5 of the line of sight from said apparatus to the target at said time, three of said mechanisms each embodying a plurality of multiplying cams each having capacity for movement in two directions diifering in kind for directly transposing one of said last three magnitudes to correspond to the respective differences in the range of E of the target from the gun, the lateral angle 0:0 and the elevational angle J30 of the line of sight from the gun to the target from the corresponding range E, lateral angle on and elevational angle of the target from said apparatus at the same time, and a plurality of means individually controlled by the multiplying cams and responsive to changes in any of the four magnitudes involved for transmitting movement reflecting said changes each to an individual cam embodying mechanism for combining such movement with one of said magnitudes, whereby said gun may be laid on said target with respect to range, lateral angle of sight and elevation.

7. An apparatus for determining the magnitudes for parallax correction required for laying an anti-aircraft gun positioned at a fixed distance therefrom for firing at a moving target overhead, said apparatus comprising four independently operatable coordinated mechanisms for the simultaneous bringing together of four magnitudes respectively representing the horizontal distance S of the gun from said apparatus, the range E of the target from said apparatus at a certain time, the lateral angle a between the lines of sight from said apparatus to the target and to the gun at the same time, and the elevational angle 5 of the line of sight from said apparatus to the target at said time, certain of said mechanisms each embodying a plurality of multiplying cams each having capacity for movement in two directions differing in kind for directly transposing said last three magnitudes to correspond to the respective differences inthe range E0 of the target from the gun, the lateral angle can and the elevational angle 50 of theline of sight from the gun to the target from the corresponding range E, lateral angle or and elevational angle ,3 of the target from said apparatus at the same time, means for periodically adjusting the magnitude representing the horizontal distance S of the gun from said apparatus to conform to changes from time to time in the position of the gun, and a plurality of means individually controlled by the multiplying cams and responsive to changes in any of the four magnitudes involved for transmitting movement reflecting said changes each to an individual cam embodying mechanism for combining such movement with one of said magnitudes, whereby said gun may be laid on said target with respect to range, lateral angle of sight and elevation irrespective of its distance from said apparatus.

8. In an apparatus for determining the magnitudes for parallax correction required for laying an anti-aircraft gun positioned at a distance therefrom comprising individua1 means for introducing data corresponding to each of four magnitudes assumed to be known, these magnitudes being the range E of the aircraft from the apparatus, the lateral angle a between the lines of sight from the apparatus to the aircraft and to the gun, and the elevational angle 5 of the line of sight from the apparatus to the aircraft, the combination of three inter-dependent cam systems coupled to said data introducing means and multiplication devices associated therewith, each of said cam systems comprising a plurality of multiplying cams, each consisting of a two-motion cam and a feeler movable relatively thereto, means for imparting two motions differing in kind to the respective cams, means for keeping the appertaining feeler in contact with each said cam, the feeler appertaining to the foremost of two successive cams being coupled to the cam of the rearmost of these two'cams, and the feeler of the rearmost ofeach such pair of cams being provided with out-put transmitting means, respectively indicating functions of the range, lateral angle of sight and elevational angle of the line of sight from the gunto the aircraft, and means controlled by said multiplying cams and responsive to changes in any of the magnitudes involved for coupling said cam systems, whereby changes'in any of said four magnitudes are refiected in the entire output of said apparatus.

9. Apparatus for obtaining simultaneous solutions of equations of the general form:

s cos a sin B AE=s cos 04 cos [3 wherein s is a fixed quantity and a, p and E are variable quantities whose magnitudes are known at any given instant, said apparatus comprising a plurality of interconnected systems of two-dimensional cams connected for operation in trains, each such cam being arranged for movement of two different kinds and shaped for producing the product of two factors, independent means for simultaneously introducing a plurality of said magnitudes individually to a corresponding number of said systems, said means serving severally for interconnecting said systems, and means cooperating with the final cam of each of said trains for receiving and transmitting the respective products constituting the solutions to said equations.

KARL PAPELLO. 

